A variational method for electromagnetic diffraction in biperiodic structures

(1)

M2AN - MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

D. C. D OBSON

A variational method for electromagnetic diffraction in biperiodic structures

M2AN - Modélisation mathématique et analyse numérique, tome 28, n^o4 (1994), p. 419-439

<http://www.numdam.org/item?id=M2AN_1994__28_4_419_0>

L’accès aux archives de la revue « M2AN - Modélisation mathématique et analyse numérique » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

(2)

A VARIATIONAL METHOD FOR ELECTROMAGNETIC DIFFRACTION IN BIPERIODIC STRUCTURES (*)

by D. C. DOBSON O

Communicated by R TEMAM

Abstract — Consider a time-harmonie electromagnetic plane wave incident on a bipenodic structure in U3 The periodic structure séparâtes two régions with constant dielectnc coefficients The dielectnc coefficient mside the structure is assumed to be a gênerai bounded measurable function The magnetic permeabihty is constant throughout IR3.

We desenbe a simple variational method for finding weak « quasipenodic » solutions to Maxwell's équations in such a structure Our formulation is simple and computationally attractive because it only involves three field components The problem is formulated by constructing a variational form over a bounded région, with « transparent » boundary conditions The boundary conditions corne from the Dinchlet-Neumann map s for the problem, which can be calculated exphcitly We show that the variational problem admits unique solutions for ail sufficiently small frequenties, and more generally for ail but a discrete set of frequencies We also show that the weak solutions satisfy a conservation of energy condition

Finally, we briefly discuss an implementation of a three-dimensional numerical finite element scheme which solves the discretized variational problem, and present the results of a simple numerical experiment

1. INTRODUCTION.

We consider a time-harmonic electromagnetic plane wave incident on a gênerai biperiodic structure in IR3. By biperiodic (or doubly periodic), we mean that the structure is periodic in each of two orthogonal directions. Such structures are sometimes called crossed diffraction gratings in the opties and physics literature. The periodic structure séparâtes two régions with constant dielectric coefficients. Inside the periodic structure, the dielectric coefficient is allowed to be a gênerai bounded measurable function. The magnetic permeability is assumed to be constant throughout the whole space.

(1) Manuscript received January, 27, 1993, revised August 30, 1993

(') Institute for Mathematics and îts Applications, University of Minnesota, Minneapolis, Minnesota, USA Present address . Department of Mathematics, Texas A & M University, College Station, Texas 77843-3368, USA

M: AN Modélisation mathématique et Analyse numérique 0764-583X/94/04/$ 4 00

(3)

4 2 0 D. C. DOBSON

This problem is motivated by applications in micro-opties, where micron- scale optie al diffraction structures are constructed with tooi s from the semiconductor industry. See the book [12] for a description of this and other mathematical problems which arise in these applications. For an introduction to the problem of electromagnetic diffraction through periodic structures (diffraction gratings), along with some numerical methods, see the in tere s t- ing collection of articles edited by R. Petit [16]. Related problems have been recently studied by Nédélec and Starling [15], Bellout and Friedman [3], and Ducomet and Quang[ll], Numerical methods for the nonperiodic time- harmonic diffraction problem have also been studied ; see Bendali [2] and the références therein.

In [10], the existence and uniqueness of solutions to Maxwell's équations in biperiodic structures, consisting of two homogeneous materials separated by a piècewise C2 interface, was established by means of an intégral équation approach. *The work generalized the earlier work of Chen and Friedman [8] and the approach was later implemented numerically [9].

Bruno and Reitich [5], [6], [7] have developed an elegant and extremely efficient method for solving diffraction problems in periodic structures, based on continuing the fields as analytic functions of the height of the interface.

The method presented hère has the advantage that it is applicable to extremely gênerai diffraction structures. In particular, the structure is allowed to be defined by a gênerai bounded measurable dielectric coefficient.

Thus there are no restrictions on the height, topology, or number of materials present in the interface.

A similar variational approach was recently taken by Abboud and Nédélec [1] to study Maxwell's équations in a bounded (nonperiodic) inhomogeneous medium. Our approach differs from [1] in that our variational formulation involves only the magnetic field vector. The approach of [1]

allows for non-constant magnetic permeability, and is thus more gênerai than our approach ; however for the optical applications which motivate the present work, the magnetic permeability is constant. For this case, our approach is somewhat simpler. Abboud and Nédélec mention in [1] that their work will be generalized to periodic structures in a fortheoming paper.

The outline of this paper is as follows. In Section 2, we describe the periodic Maxwell équations. We then dérive in Section 3 a variational form involving only the magnetic field vector in a bounded région. To complete the variational formulation of the problem requires a description of the boundary values of the magnetic field. This is obtained in Section 4 by explicitly calculating the Dirichlet-to-Neumann map for the problem, allowing us to formulate « transparent » boundary conditions. In Section 5 we then show that the variational problem has a unique solution for ail sufficiently small frequencies, and more generally for ail but a discrete set of

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis

(4)

421 frequencies. In Section 6 we show that the weak solutions satisfy a conservation of energy condition. Finally, we briefly describe a numerical finite element method based on a discretization of the variational problem and present some illustrative numerical results.

2. THE PERIODIC MAXWELL EQUATIONS

Let Lj, L2 be positive constants. Define the lattice

A = Lx Z x L2 Z x {0} c IR3 , Z = {0, ± 1, ± 2, ... } . Let b be another positive constant. Define the following domains :

Do = {x e R3 : - b < x3 < b) , D2 = {_xe ³ :

where points x e U3 are denoted x = (xl9 x2, x3). See Figure 1.

Let E, H dénote complex vector fields on IR3. The fields E, H will be called the electric and magnetic fields, respectively. We wish to solve the time-harmonic (time dependence e~lü)t) Maxwell équations

(1) (2) V x E - iwfxH = 0 ,

V x H + icosE = 0 ,

Figure 1. — Location of the three domains Z>0, Dx and D2 relative to one periodic cell. The first few lattice points n e A are indicated by dots.

(5)

422 D C DOBSON

in IR3 when the magnetic permeabihty JA IS a real constant throughout R³ and the dielectnc coefficient s is penodic with respect to A

£(x + n ) = e(x), f o r a l l n e A , x e U3,

and satisfies

f ex in a neighborhood containing D x , ' £ i n a neighborhood containing D 2 ,

where ^ and £2 a r e constants with Re (ej), Re ( £2) > 0 , lm (fj) = 0, lm O²) =2= 0 Inside D^o, s is assumed to be a bounded measurable function with Re (e)~z a > 0, lm (e ) s= 0 The case lm (e) > 0 accounts for matenals which absorb energy , see e g [4]

We will assume that a plane wave

is incident on Do from Dj Here p e IR3 is the magnetic polarization vector and q = {q^x, q², q?,) e IR³ is the incidence vector The electnc polarization s is given by s = (p x q) The vectors p and # must satisfy the

CO £ j

dispersion relation

q . q = (o2 el /A , and the orthogonahty condition

p.q = 0

in order for ( £ * , ƒ ƒ * ) to satisfy (1), (2) in D^{ We also assume that (E*, ƒƒ*) is an incoming wave, so <y³ <= 0

Let a = (qû q², 0), and define the vector fields Eâ9 Hâ by Eâ(x) = e^{ia x}E(x), Hâ(x) = e^1<X * H(x)

We are interested in quasipenodic solutions to (1), (2), that is, solutions such that the fields E^a, H^a are penodic with respect to A The Maxwell équations (1), (2) in IR³ then become

= 0 , (3) Va xHa + i<osEa = 0 , (4) where V^a = V + i a

Since all the functions we deal with henceforth are yl-periodic, to avoid continually referring to the penodic boundary conditions we instead view the M2 AN Modélisation mathématique et Analyse numenque

Mathematical Modelling and Numencal Analysis

(6)

problem as being posed over the quotient space IR^JM. We thus define the new periodic domains

n

⁰

=

D^Ö/A

,

n¹ = D^XIA, n² = D²IA , along with the boundaries

r

^l

= dn

^l

, r

²

= a/2

²

.

Note that iï⁰ is now a compact set.

For the remainder of the paper we will be studying the system (3), (4) over the quotient space U³/A. We shall henceforth drop the subscripts a from the fields Eâ and Hâ. To get solutions to the original Maxwell équations (1), (2), our solutions E and H to (3), (4) must be multiplied by eîa'^x.

3. VARIATIONAL FORM

Taking the V^a curl of l/(e^i ) times équation (4), it follows from (3), (4) that

V, x j ^ - V

^{f f}

x / / ) -co

²

H = 0, (5)

V^a x H + ia>eE = 0 . (6)

Our goal is to solve (5) for H ; if a solution exists, the field E is then formally determined by (6). It is easy to check that if H and E are differentiable then (5), (6) is equivalent to (3), (4). More generally, (5), (6) is equivalent to (3), (4) in a weak sense.

Remark : We have chosen to study the system (5), (6) rather than the

« dual » system

Va x (Va xE)- o)2 efjiE = 0 ,

- V^a x E = 0 ,

because, as can be shown from the weak form of (3), (4), the normal components of the E field across surfaces of discontinuity in e are discontinuous (see e.g. [4]). By contrast, the H field is continuous across jumps in e. Thus, given that we wish to model discontinuous £, the H field is somewhat more amenable to approximation in simple finite element spaces. Of course, the situation would be reversed if we were considering the problem with constant s and variable IA.

(7)

4 2 4 D. C. DOBSON

It is easy to check that the operator Va satisfies the usual curl and divergence identities Va x (Va u) = Va . (Va x F ) = 0. Taking the Va divergence of (5) then reveals that Va . H = 0. Therefore if H satisfies (5), it must also satisfy

Va x (yVa x H) - Va(yVa .H)-a>2H = 0, (7) where y = 1/(SJUL). Conversely, if H satisfies (7) along with the constraint Va . H = 0, then H satisfies (5).

In Section 5, we will show that (7), along with an appropriate « outgoing wave » condition, admits a unique solution H for all but a discrete set of frequencies <o. Assuming that (3), (4) also admits a solution (Ë, H) satisfying trie outgoing wave condition, the two solutions ƒƒ, H must coincide since H also solves (7). Thus under the a priori assumption of existence of solutions to the original periodic Maxwell équations (3), (4), it suffices to solve (7) ; the E field is then determined by (6). We note that the existence of solutions to the system (3), (4) is proved in [10] for structures consisting of two homogeneous materials separated by a piecewise C2 boundary.

The reason for subtracting the term Va(yVa . H) from équation (5) is to gain a coercive part in the sesquilinear form associated with the weak formulation ; this will be made clear in Section 5. We now pose the weak formulation of (7). Let F be a smooth vector field on /20. From (7) we must have that

f {V^a x (yV^a xH)} .F -

- f { . v

a

(

r

v

a

. / / ) } . F - <o

2

f

H . F = O . (8)

In équation (8), and for the remainder of the paper, bars (F, etc.) dénote complex-conjugation. Applying Green's formulas and some vector identities reveals that (8) is equivalent to

J (yVâxH). (Vâ x F ) + [ (yVâ.H)(V^7F)-co² [ H. F + + f V x {r(Vâ xH)} .F - f (rVâ.//)(F.77) = O, (9) where rj is the unit outward normal on dü0 = F{ U F2.

We wish to find a field H e Wl(nof on O0 such that (9) holds for ail

M2 AN Modélisation mathématique et Analyse numérique Mathematica! Modelling and Numerical Analysis

(8)

F e W](f}0)3. (Note : throughout the paper, given a domain D, we dénote the usual L2 Sobolev spaces HS(D) of complex-valued distributions on D by WS(D\ thus avoiding the notational conflict with the field H.) To couple the variational problem (9) to the whole space U3/A, we must find a suitable description of the derivatives of H on Sf2Q, That is the topic of the next section.

4. TRANSPARENT BOUNDARY CONDITIONS

For each index n = (nu n2, 0) e A, let

(

2 7rrci 2 7rn2 \

Since H is ^1-periodic, we expand in a Fourier series

H(x)= £ / / " ( x3) e - ' " - * , (10)

ne A

where

tf"<*3) = 7777 C H ( ; C l' *2' X3) e'""'Ar dx* dx2 '

and Q = [0, Lj] x [0, L²]. For n e /l, and y = 1, 2, define the coefficients

where

ô = a r g ( < o2 £ j n - \ an - a \2) ,

Notice that fS" is real for at most finitely many n ; for the remaining n G A, fi" has positive imaginary part. We shall assume that /?ƒ ^ 0 for ail n e A, y' = l , 2 . This assumption excludes at most a discrete set of parameters (o, /J, from considération.

Since e is constant in Op j = 1, 2, inside these régions équation (7) reduces to the periodic vector Helmholtz équation

( Aa + < o2 e jj J L ) H = 0 i n n j 9

where A^a=A + 2ia.V— \^a\²* It then follows from knowledge of the fundamental solution to the periodic Helmholtz équation (see [10]) that

(9)

426 D. C. DOBSON

inside Üx and 12 2, all fields H satisfying (7) can be represented as a sum of plane waves

" k = £ A;*-"""-1""1, 7 = 1,2 (11)

n S A

where the A* e C3 are constants. In fact, the dielectric coefficient e is constant in neighborhoods of the boundaries Fp so the représen- tation (11) is valid in a neighborhood of 12 y

Notice that only the finite number of plane waves in the sum (11) which correspond to real coefficients f3ƒ propagate outward. The remaining terms decay (or grow) exponentially as |x3| -> oo. The exponentially decaying terms are often called evanescent. We will insist that H is composed of bounded, outgoing plane waves in f2^x and Z2², plus the incoming incident wave H* = p e~l l*z in f2 v Thus we implicitly enforce an outgoing wave condition. Recalling that Fx = {x3 = b}, F2 = {x3 = - b), we obtain from matching terms in (10) and (11),

Hn(b) e ' X3 ~ , n ^ 0 , in /2 j ,

nn/v \ LJ0SL*\ lP\(x3~b) , ~lP\x3 „ ' ^ 1 ( ^ 3 - 2 0 ) ^ . -^

ri (^x3 ) = {ti \u) e + p e — p e , w = u , i n J ^ I ,

//n(— b) e 3 1 in f22 •

Thus, the fields H in the domains / 2^; have the représentation (12)

(14) From (12) we calculate the normal derivatives on the boundaries

n =*=(), y = l ,

, n = 0 , y = l , (15) 7 = 2 ,

where e3 is the unit vector in the direction of the *3-axis. It then follows formally from (10) and (15) that

OU

1 n e yl

i/3ÏH"i-b)e-

(16) (17)

(10)

1 / 2(

For functions ƒ e VK1/2(r,), define the operators TJ, j = 1, 2 by

**Ï T / = ( l 'W,*"**

¹

"

¹

**"*) ^ (18)**

where the Fourier coefficients fn are given by

JQ\ Q

and equality in (18), (19) is interpreted in the sense of distributions.

LEMMA 4.1 : The operators 7J : W^U2(r^}) - W~^m(r^j) are continuons.

Proof : TJ is an order one pseudodifferential operator (in fact, a convolution operator). See e.g. [17]. •

We see from (16), (17) that the operators IJ are « Dirichlet to Neumann maps » for the field H.

Now define the vector-valued operators /?; : Wl/2(Fj)^ W1'2^ )3 by

*? ƒ = O ? / ) ex + (3?/) ^ + (7? ƒ ) e3, M f ^ W / ) «i + O2V) e2 + (7? ƒ ) e3,

where d" = $} + iap and e} dénotes the unit vector in the x}-direction. If H s Wl(f20)3, we see that

(Va x H)\r =Rïx (H\F])- 2 ipx(-p2, Pu 0) e~lP' b ,

where again, equality is in the sense of distributions. The operators /?" thus map the components of the field H restricted to F} to their derivatives

on Fj.

Substituting the expressions above into the variational form (9) then yields the équation

(11)

4 2 8 D. C. DOBSON

f ( r V

^B

x f f ) . ( v

^B

x F ) + - f ( r V

^a

. / / ) ( v 7 7 F ) - ^

²

I / / . F

J/20 Jï2Q Jn0

+ f

- f ( y K f . / / ) ( F . *

(yKf .H) (F. e³3

) + f

) + \ {yRa2 .H) (F. e3)

= - f 2i/3ie-il3<by(p.F), (20)

J r]

where the intégrais over Fj represent the dual pairing of W~l/2(Fj) with WV2(Fj) and it is understood that the operators RJ act on the traces of//.

From (20), we define the sesquilinear form /?(//, F ) on

f, F ) = f

B(H,F)= (y Vax / / ) . ( Vax F ) -

+ f (rV

^a

.//)(v77F)-c

²

f H.

J nQ J n0

f ( r a r,O F f

J A J A

- I (y/?f . H) (F . e3) + f (y/?? . / / ) (F . e3),

J A J A

along with the functional G E [VK'C^O)3]'* defined by G(F) = - 2//?! e P l y(p.F).

JA

We then pose the variational problem : find H e Vt^/^o)3 such that

£(//, F ) = G(F), for ail

5. EXISTENCE AND UNIQUENESS OF WEAK SOLUTIONS

In this section we establish the existence and uniqueness of solutions to the variational problem (21). We show in Theorem5.1 that a unique solution exists for all frequencies 0 < a> < co 0 where <o0 is some positive number depending on the parameters Ll9 L2, e, JUL, and q. Then in Theorem 5.2 we

(12)

show that a unique solution exists for ail positive parameters a> except those in some discrete set.

THEOREM 5.1 : There exists a frequency a>0>0 such that the variational problem (21) admits a unique solution H s Wl(f20)3 for ail frequencies

0 < Ù) =S a> Q.

Proof : For the remainder of the proof, c will dénote a « generic » real constant whose value may change from line to line.

Let us write the variational form B in (21) as B(H,F) = BX(H, F ) - Ù>2B2(H, F ) where

( r Vax / / ) . ( Vax f ) + (y Va. J

a n d

+ | e³x {y(K? xH)} .F - \ e³ x {y(R% x H)} . F

- f (yR?.H){F .e3)+ \ ( y * ? . H) (F . e3) ,

Jr, Jr2

B2(H,F)= f H.F . JnQ

Applying some vector identities and integrating by parts in the xx and x2 directions reveals that

f y \VaxH\2+ f y \Va.H\

y^|a;//,|2 + 2Re (f

where the subscripts on H dénote vector components and rj3 is the j^-component of the unit outward normal y. With similar manipulations one

can show that

f v x {y(VaxH)}. H- \ (yva. /ƒ)(/ƒ. v)

dn0

(13)

4 3 0 D. C DOBSON

Thus, integrating by parts on the boundary terms and combining we obtain y(Ta2H).H. (22)

For; = 1, 2, let Af = {ne A: lm £ƒ = 0} ; define Aj = A ~ Af. The sets Af are finite and contain the indices of the outward propagating modes. The sets Aj contain the indices of the exponentially damped modes. Let us expand the terms involving the operators T* :

= - f y E iP?Hne"a*-x. H- f y £ if3»Hne~ian'x. H.

Since lm (e) =s 0, we can write y = — = y' - iy" where y ' > 0, y " ^ 0. Let us first consider the case where e2 is real. We then have from (22) that

Re {Bt(H,H)} =

= f

f y ' J M d ; / / , j2 + r i V l^'l \Hn(b)\2+y2 ^ \t3i\\Hn(~b)\\

v fin - ~

lm

= - f y ^ ( 3 ; / / , |

²

-

^{r i}

X |/8f| |/

It is easy to check that for <o sufficiently small, there is a constant c ^ 0 such that

\0?\ **CÙ>(1 4- \n\2)mf for all « e i , j = 1, 2 . f , .2 f 9

F u r t h e r m o r e , | 8" / / ^ | ^ c 13; / / ^ | and h e n c e

I^Cff, /f)| > c / £ 2; |3;//,|2 + ai ||//||ii/2(ri) + « \\H\\2wm{r2)\ . (23)

T h e c a s e w h e r e e2 h a s positive imaginary part is only a little m o r e c o m p l i c a t e d . In this case let qn = -iy2Pt and split qn into real and M2 AN Modélisation mathématique et Analyse numérique

Mathematical Modelling and Numerical Analysis

(14)

imaginary parts qn = q'n - /<^\ It is easy to check that q„ >• 0 for all n e A. Let Â = { « G v l : ^ ^ < 0 } . One can also check that A is a finite set and for all n e A, we have | ^ | > | ^ |. It follows that

We can also check that there is a constant c such that

q'^cwil + \n\2)m , for all « G A , y = 1, 2 , and so the bound (23) still holds.

Let U e W1(f20)2 be a vector-valued function with harmonie components and Dirichlet data U = H on 8/2⁰. Then (H -U)e W^{0(f2⁰) and by elliptic estimâtes (e.g. [13]),

Applying the Poincaré inequality and (23) we have

with the last inequality holding for a> sufficiently small. Thus for small (o, B^x satisfies the coercivity bound

\Bl(ƒƒ,/ƒ)! ^cx <o | | H | | ^ i( i 2 ). (24) Let us define the operator Ax : Wv(ftf -> [Wl{£lfY by (AXH,F) = BAH, F). From (24), we see that A, is invertible with IIA7^x II =s .

11 " Cj û>

Notice also that the operator A2 : ^ ( / ï )3 -• [Wl(O )3] ' defined by

(15)

4 3 2 D. C. DOBSON

(A²H,F) = B²(H, F) is bounded (and in fact, compact). Defining A = A, — Ù>2A2 and bounding IIA\ l A2\\ =s we then have

Cj O)

1 — a) \AX A21 | cx <o — c2 <o 2

Hence for a> sufficiently small, A"¹ exists. D

THEOREM 5.2 : The variational problem (21) admits a unique solution H e Wl(f2Q)3 for ail but a discrete set of parameters co > 0.

Proof : Let M = {co : p" = a>2 e. ix - \ an - a |2 ^ 0 for ail

j K ' j j * i n i r

n e A , / = 1 , 2 } , so M is a discrete set of points. Recall that we excluded parameters such that co e M in Section 4 when we defined the boundary operators T".

Note that the sesquilinear form B^x dépends on co through the coefficients P* defining the operators T". Nevertheless, for any fixed co £ 0$, we can bound

\P1\ s s c ( l + \n\2)m , for ail n e A , y = 1, 2 ,

and so with the same argument used in Theorem 5.1 we can establish a bound

so that the associated operator A j is bounded and invertible. To emphasize that Ax dépends on o>, we write A1(w). The compact operator A2 does not depend on a>.

Holding a>x $ M fixed, now consider the operator A (w1? <u) = A1( ( y1) - ct>2A2. By Fredholm theory, we see that A(Ù>^X, W)'¹ exists for ail

<o ^ ^(o>i), where ^ ( w ! ) is some discrete set. We will first show that A(o>, (o)~l exists for <o near <wl9 except possibly for <o e ^ ( w j ) , thus

proving the theorem for a> near wp It is easy to check that

H A ^ W ) - A ^ O H ^ 0 , as (o -+ (o 2 ;

indeed, one need only show that the boundary maps T"{CÙ), T°(<OX)

converge as operators on W^1/2(F) and this follows from the définition of the c o e f f i c i e n t s P". T h u s , s i n c e | | A ( o > , o>) — A(<ox, <o)\\ = \\Ax(a)) - Ax(a>x)\\

is small for | co - &>^x \ sufficiently small, it follows from the stability of bounded invertibility (see e.g. [14], Theorem IV-1.16) thatA(o>, a> )" 1 exists and is bounded for ail co near co^x with a> $ l ( w , ) .

(16)

Since a>^x can be taken at any point outside ^ , it follows that A((o, (o)~l exists for all but a discrete set of frequencies (o, •

6. CONSERVATION OF ENERGY

In this section we show that the weak solutions of the variational problem(21) conserve energy in the sense that the energy radiated away from H^o is the same as that of the incident wave, provided there is no material present which absorbs energy (i.e. provided lm (e) = 0).

From the expansion (11) and the représentation (13), we see that the coefficients of each propagating reflected plane wave are

rn = Hn(b)e~l0lh for n ^ O , n e Af ,

~l^b~pe-2l^b for n = 0 ,

where Af = ( n e A : lm £ " = 0 } . The « energy » of each reflected mode can be defined [16] by /?" \rn\2//31. (Note : there are many different ways to define the energy of electromagnetic waves. Our définition is equivalent, up to a multiplicative constant, to the usual définition involving the Poynting vector.) Similar to the reflected modes, the coefficients of each propagating transmitted more are

tm=Hm(-b)e~ll*lb for m e A2+ ,

where A2 = {m e A : lm /3 ™ = 0} . Note that propagating transmitted modes only exist when y2 = l/(At e2) is real, for otherwise A% is empty. The energy of each transmitted mode is similarly defined as

, m e Ai .

T / »

Conservation of energy states that the total energy of the reflected and transmitted waves must equal the energy of the incident wave ƒƒ*. The energy of the incident wave is |/?|2, where p is the magnetic polarization vector. The conservation of energy condition can thus be written

^ ( l ^"\

r

"\

2 +

y 1 e ? K \

2

) = \P\

2

•

THEOREM 6.1 : (Conservation of energy). Assume that the coefficients y, yu and y2 are real. Then the reflection ans transmission coefficients rn, tm corresponding to solutions H e Wx(üf of (21) satisfy (25).

(17)

4 3 4 D. C. DOBSON

Proof : From (21) we see that

, H) = - 2 ^ ! Re

{• ¹ *'J>o-">}-

Expanding the operators 7?, T^ we see that the imaginary part of i? is given by

- n £ /?n/ I

n e A* m e A%

= - 2 / 3 ,r iR e from which (25) follows immediately. D

Remark: Scaling the incident wave energy to | p |2= 1, the numbers (0? k«|2V0i a n d ( r2 021 Um|2)/(yi 0 i ) are often referred to in the opties and engineering literature as efficiënties. They represent the proportion of energy radiated in each propagating mode. Given a solution H to prob- lem(21), the efficiencies can be easily calculated from the Fourier coef- ficients of the traces H | r .

7. FINITE ELEMENT METHOD AND NUMERICAL EXPERIMENTS

In this section, we briefly describe a finite element method for solving the variational problem (21). We will not prove any convergence theorems or make error estimâtes here. Such estimâtes should be facilitated by the fact that the underlying operator A has a bounded inverse, as established in Theorems 5.1 and 5.2, although in the case of rough dielectric coefficients, the lack of regularity may preclude « optimal » global error esimates for standard schemes. Rather than delve into these issues here, we merely wish to describe a particular finite element scheme and illustrate qualitatively, with a simple numerical experiment, the kinds of results one can expect with such a scheme.

It is very convenient that the domain 120 is a periodic «box». We discretized f20 with a uniform rectangular grid, say with Nk grid points in each xk direction, and used the piecewise-trilinear finite element basis with éléments <f> (x - XJ\ where {XJ} is the set of Nx . N2. Af3 grid points, and

otherwise .

(18)

ELECTROMAGNETIC DIFFRACTION 435 Hère, h^k dénotes the grid spacing in the A>th direction, Le., h^x =L¹/N¹, h2 = L2/N2, h3 = 2 b/(N3 - 1 ). The basis éléments centered at grid points which lie on the boundaries Fj are restricted to O0.

The équations were discretized using the standard Galerkin approximation with a simple gaussian quadrature formula to compute the intégrais over the cells. The boundary conditions were calculated by truncating the Fourier transform représentation of the operators TJ.

The resulting linear System is large and sparse, but not necessarily Hermitian or positive definite. We chose to solve the system with an

« Orthomin » itérative solver. We experimented with several generic preconditioners, for example diagonal scaling and incomplete LU décomposition, but generally observed quicker convergence without a preconditioner. An important topic for future research is the design of effective preconditioners for this problem.

Our numerical experiment is meant to solve a prototypical electromagnetic scattering problem which could be encountered in micro-optics. We are given a biperiodic structure such as the one pictured in Figure 2. This particular structure illustrâtes qualitatively the features of devices we wish to model.

The goal is to predict the diffracted field which results when a plane wave of a particular frequency, polarization, and incidence angle impinges on the

X3

- UI Aft

Figure 2. — Cross-section through the center of one periodic cell of the prototypical biperiodic structure. The profile is the same in both the (xv xj and (x2, x3) cross-sections.

(19)

436 D. C. DOBSON

structure from above. In micro-optical applications, the period of the structure is usually comparable to the wavelength of the incident radiation.

We have chosen the wavelength of the incoming wave to be 0.55 |xm, roughly in the center of the visible electromagnetic spectrum. We have taken the period of the structure to be 0.5 |xm in both the xx and x2-directions. The

1.5 1

Figure 3. — Cross-section of the amplitude | H | , taken through the métal région in the (*Ü*3) Plane.

2 1.5

1 0 . 5

10 x 3 - a x i s

15 x l - a x i s

Figure 4. — Cross-section of the amplitude | H | , taken through the non-absorbtive région in the (*!, JT3) plane.

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analyste

(20)

ELECTROMAGNETIC DIFFRACTION 437 height of the computational box is also 0.5 jxm. We used a 32 x 32 x 33 grid. The grid line s are aligned with the discontinuities in the dielectric coefficient e, since this is where we expect discontinuities in the derivatives of//.

Figures 3 through 6 show cross-sections of the amplitude of the H field inside the computational box when an incoming plane wave,

2

1.5 1

s— «zr__ *-•—-

Figure 5. — Cross-section of the amplitude \H\, taken through the métal région in the Ui,*²) plane.

30

10 x 2 - a x i s

15 x l - a x i s

Figure 6. — Cross-section of the amplitude | / / | , taken below the métal région in the (xvx2) plane.

(21)

438 D. C DOBSON

polarized such that the E field is pointing in the x2 direction, is incident on the structure at an angle of 30° with respect to the x3-axis. Perhaps the most striking feature of the figures is the sharp drop in field intensity inside the metallic material (which we chose to be gold), compared to the relatively smooth « waves » in the other media. Notice that the intensity is symmetrie in the x2-direction, but asymmetrie in the xrdirection ; this is because the incoming wavevector is orthogonal to the x2-axis.

The primary advantages of this scheme are its generality and simplicity.

Perhaps the biggest disadvantage at present is the large cost associated with complicated simulations. In gênerai, as the ratio of the wavelength to the size of the structure decreases, the waves inside the box become more complicated and therefore more difficult to approximate accurately in finite element spaces. Since we must use a 3-dimensional mesh, the cost of adding more grid points in each direction is large. As it stands, the method is certainly feasible for « medium sized » problems (say 20 x 20 x 20 mesh) on the current génération of werkstations. Hopefully, with improvements in preconditioners and itérative methods for solving the discretized équations, the cost can be reduced significantly.

ACKNOWLEDGEMENTS

I wish to thank Dr. J. Allen Cox from Honeywell, Inc. for introducing me to this problem and some of its applications. Thanks also to Prof. Anne Greenbaum, who suppiied the Orthomin code used in the numerical implementation. The author gratefully acknowledges the support of the Minnesota Supercomputer Institute, which provided Cray-2 time and excellent technical assistance for the numerical experiments. Finally, thanks to the référée for several helpful suggestions and comments.

REFERENCES

[1] T. ABBOUD, J. NÉDÉLEC, 1992, Electromagnetic waves in an inhomogeneous medium, J, Math. Anal. AppL, 164, 40-58.

[2] A. BENDALI, 1984, Numerical analysis of the exterior boundary value problem for the time-harmonic Maxwell équations by a boundary finite element method.

Part 1 : The continuous problem. Part 2 : The discrete problem, Math. of Computation, 167, 29-46 and 47-68.

[3] H. BELLOUT, A. FRIEDMAN, 1990, Scattering by stripe grating, / . Math. Anal.

AppL, 147, 228-248.

[4] M. BORN, E. WOLF, 1980, Principles of Opties, sixth édition, Pergamon Press, Oxford.

[5] O. P. BRUNO, F. REITICH, 1992, Solution of a boundary value problem for Helmholtz équation via variation of the boundary into the complex domain, Proc. Royal Soc. Edinburgh, 122A, 317-340.

(22)

[6] O. P. BRUNO, F. REITICH, 1993, Numerical solution of diffraction problems : a method of variation of boundaries, / . Opt. Soc. America A, 10, 1168-1175.

[7] O. P. BRUNO, F. REITICH, Numerical solution of diffraction problems : a method of variation of boundaries II. Dielectric gratings, Padé approximants and singularities ; III. Doubly periodic gratings, preprints.

[8] X. CHEN, A. FRIEDMAN, 1991, MaxwelFs équations in a periodic structure, Trans. Amer. Math. Soc, 323, 465-507.

[9] J. A. Cox, D. DOBSON, 1991, An intégral équation method for biperiodic diffraction structures, in J. Lerner and W. McKinney, éd., International Conference on the Application and Theory of Periodic Structures, Proc. SPIE 1545, 106-113.

[10] D. DOBSON, A. FRIEDMAN, 1992, The time-harmonic Maxwell équations in a doubly periodic structure, / . Math. Anal AppL, 166, 507-528.

[11] B. DUCOMET, D. Ha. QUANG, 1992, Diffusion électromagnétique à basse fréquence par un réseau de cylindres diélectriques : étude numérique, RAIRO, ModéL Math. Anal. Numér. 26, 709-738.

[12] A. FRIEDMAN, 1990, Mathematics in Industrial Problems, Part 3, Springer- Verlag, Heidelberg.

[13] D. GILBARG, N. S. TRUDINGER, 1977, Elliptic Partial Dijferential Equations of Second Order, Springer-Verlag, Heidelberg.

[14] T. KATO, 1980, Perturbation Theory for Linear Operators (corrected second édition), Springer-Verlag, Berlin.

[15] J. C. NÉDÉLEC, F. STARLING, 1988, Intégral équation methods in quasi- periodic diffraction problems for the time-harmonic Maxwell's équations, in

« Rapport Interne », Vol. 179, C.M.A.P., Ecole Polytechnique, Palaiseau.

[16] Electromagnetic Theory of Gratings, 1980, Topics in Current Physics, Vol. 22, edited by R. Petit, Springer-Verlag, Heidelberg.

[17] M. TAYLOR, 1981, Pseudodifferential Operators, Princeton University Press, Princeton, N. J.

A variational method for electromagnetic diffraction in biperiodic structures

D. C. D OBSON